Monads for functional programming

Transcription

1 Monads for functional programming Philip Wadler, University of Glasgow Department of Computing Science, University of Glasgow, G12 8QQ, Scotland Abstract. The use of monads to structure functional programs is described. Monads provide a convenient framework for simulating effects found in other languages, such as global state, exception handling, output, or non-determinism. Three case studies are looked at in detail: how monads ease the modification of a simple evaluator; how monads act as the basis of a datatype of arrays subject to in-place update; and how monads can be used to build parsers. 1 Introduction Shall I be pure or impure? The functional programming community divides into two camps. Pure languages, such as Miranda 0 and Haskell, are lambda calculus pure and simple. Impure languages, such as Scheme and Standard ML, augment lambda calculus with a number of possible effects, such as assignment, exceptions, or continuations. Pure languages are easier to reason about and may benefit from lazy evaluation, while impure languages offer efficiency benefits and sometimes make possible a more compact mode of expression. Recent advances in theoretical computing science, notably in the areas of type theory and category theory, have suggested new approaches that may integrate the benefits of the pure and impure schools. These notes describe one, the use of monads to integrate impure effects into pure functional languages. The concept of a monad, which arises from category theory, has been applied by Moggi to structure the denotational semantics of programming languages [13, 14]. The same technique can be applied to structure functional programs [21, 23]. The applications of monads will be illustrated with three case studies. Section 2 introduces monads by showing how they can be used to structure a simple evaluator so that it is easy to modify. Section 3 describes the laws satisfied by Appears in: J. Jeuring and E. Meijer, editors, Advanced Functional Programming, Proceedings of the Båstad Spring School, May 1995, Springer Verlag Lecture Notes in Computer Science 925. A previous version of this note appeared in: M. Broy, editor, Program Design Calculi, Proceedings of the Marktoberdorf Summer School, 30 July 8 August Some errata fixed August 2001; thanks to Dan Friedman for pointing these out. 0 Miranda is a trademark of Research Software Limited.

2 monads. Section 4 shows how monads provide a new solution to the old problem of providing updatable state in pure functional languages. Section 5 applies monads to the problem of building recursive descent parsers; this is of interest in its own right, and because it provides a paradigm for sequencing and alternation, two of the central concepts of computing. It is doubtful that the structuring methods presented here would have been discovered without the insight afforded by category theory. But once discovered they are easily expressed without any reference to things categorical. No knowledge of category theory is required to read these notes. The examples will be given in Haskell, but no knowledge of that is required either. What is required is a passing familiarity with the basics of pure and impure functional programming; for general background see [3, 12]. The languages refered to are Haskell [4], Miranda [20], Standard ML [11], and Scheme [17]. 2 Evaluating monads Pure functional languages have this advantage: all flow of data is made explicit. And this disadvantage: sometimes it is painfully explicit. A program in a pure functional language is written as a set of equations. Explicit data flow ensures that the value of an expression depends only on its free variables. Hence substitution of equals for equals is always valid, making such programs especially easy to reason about. Explicit data flow also ensures that the order of computation is irrelevant, making such programs susceptible to lazy evaluation. It is with regard to modularity that explicit data flow becomes both a blessing and a curse. On the one hand, it is the ultimate in modularity. All data in and all data out are rendered manifest and accessible, providing a maximum of flexibility. On the other hand, it is the nadir of modularity. The essence of an algorithm can become buried under the plumbing required to carry data from its point of creation to its point of use. Say I write an evaluator in a pure functional language. To add error handling to it, I need to modify each recursive call to check for and handle errors appropriately. Had I used an impure language with exceptions, no such restructuring would be needed. To add a count of operations performed to it, I need to modify each recursive call to pass around such counts appropriately. Had I used an impure language with a global variable that could be incremented, no such restructuring would be needed. To add an execution trace to it, I need to modify each recursive call to pass around such traces appropriately. Had I used an impure language that performed output as a side effect, no such restructuring would be needed. Or I could use a monad. These notes show how to use monads to structure an evaluator so that the changes mentioned above are simple to make. In each case, all that is required is to redefine the monad and to make a few local changes.

3 This programming style regains some of the flexibility provided by various features of impure languages. It also may apply when there is no corresponding impure feature. It does not eliminate the tension between the flexibility afforded by explicit data and the brevity afforded by implicit plumbing; but it does ameliorate it to some extent. The technique applies not just to evaluators, but to a wide range of functional programs. For a number of years, Glasgow has been involved in constructing a compiler for the functional language Haskell. The compiler is itself written in Haskell, and uses the structuring technique described here. Though this paper describes the use of monads in a program tens of lines long, we have experience of using them in a program three orders of magnitude larger. We begin with the basic evaluator for simple terms, then consider variations that mimic exceptions, state, and output. We analyse these for commonalities, and abstract from these the concept of a monad. We then show how each of the variations fits into the monadic framework. 2.1 Variation zero: The basic evaluator The evaluator acts on terms, which for purposes of illustration have been taken to be excessively simple. data Term = Con Int Div Term Term A term is either a constant Con a, where a is an integer, or a quotient, Div t u, where t and u are terms. The basic evaluator is simplicity itself. eval :: Term Int eval (Con a) = a eval (Div t u) = eval t eval u The function eval takes a term to an integer. If the term is a constant, the constant is returned. If the term is a quotient, its subterms are evaluated and the quotient computed. We use to denote integer division. The following will provide running examples. answer, error :: Term answer = (Div (Div (Con 1972 ) (Con 2 )) (Con 23 )) error = (Div (Con 1 ) (Con 0 )) Computing eval answer yields the value of (( ) 23 ), which is 42. The basic evaluator does not incorporate error handling, so the result of eval error is undefined. 2.2 Variation one: Exceptions Say it is desired to add error checking, so that the second example above returns a sensible error message. In an impure language, this is easily achieved with the use of exceptions.

4 In a pure language, exception handling may be mimicked by introducing a type to represent computations that may raise an exception. data M a = Raise Exception Return a type Exception = String A value of type M a either has the form Raise e, where e is an exception, or Return a, where a is a value of type a. By convention, a will be used both as a type variable, as in M a, and as a variable ranging over values of that type, as in Return a. (A word on the difference between data and type declarations. A data declaration introduces a new data type, in this case M, and new constructors for values of that type, in this case Raise and Return. A type declaration introduces a new name for an existing type, in this case Exception becomes another name for String.) It is straightforward, but tedious, to adapt the evaluator to this representation. eval :: Term M Int eval (Con a) = Return a eval (Div t u) = case eval t of Raise e Raise e Return a case eval u of Raise e Raise e Return b if b = 0 then Raise divide by zero else Return (a b) At each call of the evaluator, the form of the result must be checked: if an exception was raised it is re-raised, and if a value was returned it is processed. Applying the new evaluator to answer yields (Return 42 ), while applying it to error yields (Raise divide by zero ). 2.3 Variation two: State Forgetting errors for the moment, say it is desired to count the number of divisions performed during evaluation. In an impure language, this is easily achieved by the use of state. Set a given variable to zero initially, and increment it by one each time a division occurs. In a pure language, state may be mimicked by introducing a type to represent computations that act on state. type M a = State (a, State) type State = Int Now a value of type M a is a function that accepts the initial state, and returns the computed value paired with the final state.

5 Again, it is straightforward but tedious to adapt the evaluator to this representation. eval :: Term M Int eval (Con a) x = (a, x) eval (Div t u) x = let (a, y) = eval t x in let (b, z) = eval u y in (a b, z + 1 ) At each call of the evaluator, the old state must be passed in, and the new state extracted from the result and passed on appropriately. Computing eval answer 0 yields (42, 2 ), so with initial state 0 the answer is 42 and the final state is 2, indicating that two divisions were performed. 2.4 Variation three: Output Finally, say it is desired to display a trace of execution. In an impure language, this is easily done by inserting output commands at appropriate points. In a pure language, output may be mimicked by introducing a type to represent computations that generate output. type M a = (Output, a) type Output = String Now a value of type M a consists of the output generated paired with the value computed. Yet again, it is straightforward but tedious to adapt the evaluator to this representation. eval :: Term M Int eval (Con a) = (line (Con a) a, a) eval (Div t u) = let (x, a) = eval t in let (y, b) = eval u in (x + y + line (Div t u) (a b), a b) line line t a :: Term Int Output = eval ( + showterm t + ) + showint a + At each call of the evaluator, the outputs must be collected and assembled to form the output of the enclosing call. The function line generates one line of the output. Here showterm and showint convert terms and integers to strings, + concatenates strings, and represents the string consisting of a newline. Computing eval answer returns the pair (x, 42 ), where x is the string eval (Con 1972 ) 1972 eval (Con 2 ) 2 eval (Div (Con 1972 ) (Con 2 )) 986 eval (Con 23 ) 23 eval (Div (Div (Con 1972 ) (Con 2 )) (Con 23 ))) 42

6 which represents a trace of the computation. From the discussion so far, it may appear that programs in impure languages are easier to modify than those in pure languages. But sometimes the reverse is true. Say that it was desired to modify the previous program to display the execution trace in the reverse order: eval (Div (Div (Con 1972 ) (Con 2 )) (Con 23 ))) 42 eval (Con 23 ) 23 eval (Div (Con 1972 ) (Con 2 )) 986 eval (Con 2 ) 2 eval (Con 1972 ) 1972 This is simplicity itself to achieve with the pure program: just replace the term with the term x + y + line (Div t u) (a b) line (Div t u) (a b) + y + x. It is not so easy to modify the impure program to achieve this effect. The problem is that output occurs as a side-effect of computation, but one now desires to display the result of computing a term before displaying the output generated by that computation. This can be achieved in a variety of ways, but all require substantial modification to the impure program. 2.5 A monadic evaluator Each of the variations on the interpreter has a similar structure, which may be abstracted to yield the notion of a monad. In each variation, we introduced a type of computations. Respectively, M represented computations that could raise exceptions, act on state, and generate output. By now the reader will have guessed that M stands for monad. The original evaluator has the type Term Int, while in each variation its type took the form Term M Int. In general, a function of type a b is replaced by a function of type a M b. This can be read as a function that accepts an argument of type a and returns a result of type b, with a possible additional effect captured by M. This effect may be to act on state, generate output, raise an exception, or what have you. What sort of operations are required on the type M? Examination of the examples reveals two. First, we need a way to turn a value into the computation that returns that value and does nothing else. unit :: a M a Second, we need a way to apply a function of type a M b to a computation of type M a. It is convenient to write these in an order where the argument comes before the function. ( ) :: M a (a M b) M b

7 A monad is a triple (M, unit, ) consisting of a type constructor M and two operations of the given polymorphic types. These operations must satisfy three laws given in Section 3. We will often write expressions in the form m λa. n where m and n are expressions, and a is a variable. The form λa. n is a lambda expression, with the scope of the bound variable a being the expression n. The above can be read as follows: perform computation m, bind a to the resulting value, and then perform computation n. Types provide a useful guide. From the type of ( ), we can see that expression m has type M a, variable a has type a, expression n has type M b, lambda expression λa. n has type a M b, and the whole expression has type M b. The above is analogous to the expression let a = m in n. In an impure language, this has the same reading: perform computation m, bind a to the resulting value, then perform computation n and return its value. Here the types say nothing to distinguish values from computations: expression m has type a, variable a has type a, expression n has type b, and the whole expression has type b. The analogy with let explains the choice of the order of arguments to. It is convenient for argument m to appear before the function λa. n, since computation m is performed before computation n. The evaluator is easily rewritten in terms of these abstractions. eval :: Term M Int eval (Con a) = unit a eval (Div t u) = eval t λa. eval u λb. unit (a b) A word on precedence: lambda abstraction binds least tightly and application binds most tightly, so the last equation is equivalent to the following. eval (Div t u) = ((eval t) (λa. ((eval u) (λb. (unit (a b)))))) The type Term M Int indicates that the evaluator takes a term and performs a computation yielding an integer. To compute (Con a), just return a. To compute (Div t u), first compute t, bind a to the result, then compute u, bind b to the result, and then return a b. The new evaluator is a little more complex than the original basic evaluator, but it is much more flexible. Each of the variations given above may be achieved by simply changing the definitions of M, unit, and, and by making one or two local modifications. It is no longer necessary to re-write the entire evaluator to achieve these simple changes.

8 2.6 Variation zero, revisited: The basic evaluator In the simplest monad, a computation is no different from a value. type M a = a unit unit a ( ) a k :: a I a = a :: M a (a M b) M b = k a This is called the identity monad: M is the identity function on types, unit is the identity function, and is just application. Taking M, unit, and as above in the monadic evaluator of Section 2.5 and simplifying yields the basic evaluator of Section Variation one, revisited: Exceptions In the exception monad, a computation may either raise an exception or return a value. data M a = Raise Exception Return a type Exception = String unit unit a ( ) m k raise raise e :: a M a = Return a :: M a (a M b) M b = case m of Raise e Raise e Return a k a :: Exception M a = Raise e The call unit a simply returns the value a. The call m k examines the result of the computation m: if it is an exception it is re-raised, otherwise the function k is applied to the value returned. Just as in the identity monad is function application, in the exception monad may be considered as a form of strict function application. Finally, the call raise e raises the exception e. To add error handling to the monadic evaluator, take the monad as above. Then just replace unit (a b) by if b = 0 then raise divide by zero else unit (a b) This is commensurate with change required in an impure language. As one might expect, this evaluator is equivalent to the evaluator with exceptions of Section 2.2. In particular, unfolding the definitions of unit and in this section and simplifying yields the evaluator of that section.

9 2.8 Variation two, revisited: State In the state monad, a computation accepts an initial state and returns a value paired with the final state. type M a = State (a, State) type State = Int unit :: a M a unit a = λx. (a, x) ( ) :: M a (a M b) M b m k = λx. let (a, y) = m x in let (b, z) = k a y in (b, z) tick :: M () tick = λx. ((), x + 1 ) The call unit a returns the computation that accept initial state x and returns value a and final state x; that is, it returns a and leaves the state unchanged. The call m k performs computation m in the initial state x, yielding value a and intermediate state y; then performs computation k a in state y, yielding value b and final state z. The call tick increments the state, and returns the empty value (), whose type is also written (). In an impure language, an operation like tick would be represented by a function of type () (). The spurious argument () is required to delay the effect until the function is applied, and since the output type is () one may guess that the function s purpose lies in a side effect. In contrast, here tick has type M (): no spurious argument is needed, and the appearance of M explicitly indicates what sort of effect may occur. To add execution counts to the monadic evaluator, take the monad as above. Then just replace unit (a b) by tick λ(). unit (a b) (Here λ. e is equivalent to λx. e where x :: () is some fresh variable that does not appear in e; it indicates that the value bound by the lambda expression must be ().) Again, this is commensurate with change required in an impure language. Simplifying yields the evaluator with state of Section 2.3.

10 2.9 Variation three, revisited: Output In the output monad, a computation consists of the output generated paired with the value returned. type M a = (Output, a) type Output = String unit :: a M a unit a = (, a) ( ) m k :: M a (a M b) M b = let (x, a) = m in let (y, b) = k a in (x + y, b) out :: Output M () out x = (x, ()) The call unit a returns no output paired with a. The call m k extracts output x and value a from computation m, then extracts output y and value b from computation k a, and returns the output formed by concatenating x and y paired with the value b. The call out x returns the computation with output x and empty value (). To add execution traces to the monadic evaluator, take the monad as above. Then in the clause for Con a replace unit a by out (line (Con a) a) λ(). unit a and in the clause for Div t u replace unit (a b) by out (line (Div t u) (a b)) λ(). unit (a b) Yet again, this is commensurate with change required in an impure language. Simplifying yields the evaluator with output of Section 2.4. To get the output in the reverse order, all that is required is to change the definition of, replacing x + y by y + x. This is commensurate with the change required in the pure program, and rather simpler than the change required in an impure program. You might think that one difference between the pure and impure versions is that the impure version displays output as it computes, while the pure version will display nothing until the entire computation completes. In fact, if the pure language is lazy then output will be displayed in an incremental fashion as the computation occurs. Furthermore, this will also happen if the order of output is reversed, which is much more difficult to arrange in an impure language. Indeed, the easiest way to arrange it may be to simulate lazy evaluation. 3 Monad laws The operations of a monad satisfy three laws.

11 Left unit. Compute the value a, bind b to the result, and compute n. The result is the same as n with value a substituted for variable b. unit a λb. n = n[a/b]. Right unit. Compute m, bind the result to a, and return a. The result is the same as m. m λa. unit a = m. Associative. Compute m, bind the result to a, compute n, bind the result to b, compute o. The order of parentheses in such a computation is irrelevant. m (λa. n λb. o) = (m λa. n) λb. o. The scope of the variable a includes o on the left but excludes o on the right, so this law is valid only when a does not appear free in o. A binary operation with left and right unit that is associative is called a monoid. A monad differs from a monoid in that the right operand involves a binding operation. To demonstrate the utility of these laws, we prove that addition is associative. Consider a variant of the evaluator based on addition rather than division. data Term = Con Int Add Term Term eval :: Term M Int eval (Con a) = unit a eval (Add t u) = eval t λa. eval u λb. unit (a b) We show that evaluation of Add t (Add u v) and Add (Add t u) v, both compute the same result. Simplify the left term: eval (Add t (Add u v)) = { def n eval } eval t λa. eval (Add u v) λx. unit (a + x) = { def n eval } eval t λa. (eval u λb. eval v λc. unit (b + c)) λx. unit (a + x) = { associative } eval t λa. eval u λb. eval v λc. unit (b + c) λx. unit (a + x) = { left unit } eval t λa. eval u λb. eval v λc. unit (a + (b + c)) Simplify the right term similarly: eval (Add (Add t u) v) = { as before } eval t λa. eval u λb. eval v λc. unit ((a + b) + c)

12 The result follows by the associativity of addition. This proof is trivial; without the monad laws, it would be impossible. The proof works in any monad: exception, state, output. This assumes that the code is as above: if it is modified then the proof also must be modified. Section 2.3 modified the program by adding calls to tick. In this case, associativity still holds, as can be demonstrated using the law tick λ(). m = m λ(). tick which holds so long at tick is the only action on state within m. Section 2.4 modified the program by adding calls to line. In this case, the addition is no longer associative, in the sense that changing parentheses will change the trace, though the computations will still yield the same value. As another example, note that for each monad we can define the following operations. map :: (a b) (M a M b) map f m = m λa. unit (f a) join join z :: M (M a) M a = z λm. m The map operation simply applies a function to the result yielded by a computation. To compute map f m, first compute m, bind a to the result, and then return f a. The join operation is trickier. Let z be a computation that itself yields a computation. To compute join z, first compute z, binds m to the result, and then behaves as computation m. Thus, join flattens a mind-boggling double layer of computation into a run-of-the-mill single layer of computation. As we will see in Section 5.1, lists form a monad, and for this monad map applies a function to each element of a list, and join concatenates a list of lists. Using id for the identity function (id x = x), and ( ) for function composition ((f g) x = f (g x)), one can then formulate a number of laws. map id map (f g) = id = map f map g map f unit = unit f map f join = join map (map f ) join unit = id join map unit = id join map join = join join m k = join (map k m) The proof of each is a simple consequence of the definitions of map and join and the three monad laws. Often, monads are defined not in terms of unit and, but rather in terms of unit, join, and map [10, 13]. The three monad laws are replaced by the first seven of the eight laws above. If one defines by the eighth law, then the three monad laws follow. Hence the two definitions are equivalent.

13 4 State Arrays play a central role in computing because they closely match current architectures. Programs are littered with array lookups such as x[i] and array updates such as x[i] := v. These operations are popular because array lookup is implemented by a single indexed fetch instruction, and array update by a single indexed store. It is easy to add arrays to a functional language, and easy to provide efficient array lookup. How to provide efficient array update, on the other hand, is a question with a long history. Monads provide a new answer to this old question. Another question with a long history is whether it is desirable to base programs on array update. Since so much effort has gone into developing algorithms and architectures based on arrays, we will sidestep this debate and simply assume the answer is yes. There is an important difference between the way monads are used in the previous section and the way monads are used here. The previous section showed monads help to use existing language features more effectively; this section shows how monads can help define new language features. No change to the programming language is required, but the implementation must provide a new abstract data type, perhaps as part of the standard prelude. Here monads are used to manipulate state internal to the program, but the same techniques can be use to manipulate extenal state: to perform input/output, or to communicate with other programming languages. The Glasgow implementation of Haskell uses a design based on monads to provide input/output and interlanguage working with the imperative programming language C [15]. This design has been adopted for version 1.3 of the Haskell standard. 4.1 Arrays Let Arr be the type of arrays taking indexes of type Ix and yielding values of type Val. The key operations on this type are newarray :: Val Arr, index :: Ix Arr Val, update :: Ix Val Arr Arr. The call newarray v returns an array with all entries set to v; the call index i x returns the value at index i in array x; and the call update i v x returns an array where index i has value v and the remainder is identical to x. The behaviour of these operations is specified by the laws index i (newarray v) = v, index i (update i v x) = v, index i (update j v x) = index i x, if i j.

14 In practice, these operations would be more complex; one needs a way to specify the index bounds, for instance. But the above suffices to explicate the main points. The efficient way to implement the update operation is to overwrite the specified entry of the array, but in a pure functional language this is only safe if there are no other pointers to the array extant when the update operation is performed. An array satisfying this property is called single threaded, following Schmidt [18]. Consider building an interpreter for a simple imperative language. The abstract syntax for this language is represented by the following data types. data Term = Var Id Con Int Add Term Term data Comm = Asgn Id Term Seq Comm Comm If Term Comm Comm data Prog = Prog Comm Term Here Id : baastad.tex, v /02 /2915 : 17 : 01wadlerExp is an unspecified type of identifiers. A term is a variable, a constant, or the sum of two terms; a command is an assignment, a sequence of two commands, or a conditional; and a program consists of a command followed by a term. The current state of execution will be modelled by an array where the indexes are identifiers and the corresponding values are integers. Here is the interpreter. type State = Arr type Ix = Id type Val = Int eval :: Term State Int eval (Var i) x = index i x eval (Con a) x = a eval (Add t u) x = eval t x + eval u x exec :: Comm State State exec (Asgn i t) x = update i (eval t x) x exec (Seq c d) x = exec d (exec c x) exec (If t c d) x = if eval t x = 0 then exec c x else exec d x elab :: Prog Int elab (Prog c t) = eval t (exec c (newarray 0 )) This closely resembles a denotational semantics. The evaluator for terms takes a term and a state and returns an integer. Evaluation of a variable is implemented by indexing the state. The executor for commands takes a command and the initial state and returns the final state. Assignment is implemented by updating the state. The elaborator for programs takes a program and returns an integer. It executes the command in an initial state where all identifiers map to 0, then evaluates the given expression in the resulting state and returns its value.

15 The state in this interpreter is single threaded: at any moment of execution there is only one pointer to the state, so it is safe to update the state in place. In order for this to work, the update operation must evaluate the new value before placing it in the array. Otherwise, the array may contain a closure that itself contains a pointer to the array, violating the single threading property. In semantic terms, one says that update is strict in all three of its arguments. A number of researchers have proposed analyses that determine whether a given functional program uses an array in a single threaded manner, with the intent of incorporating such an analysis into an optimising compiler. Most of these analyses are based on abstract interpretation [1]. Although there has been some success in this area, the analyses tend to be so expensive as to be intractable [2, 7]. Even if such analyses were practicable, their use may be unwise. Optimising update can affect a program s time and space usage by an order of magnitude or more. The programmer must be assured that such an optimisation will occur in order to know that the program will run adequately fast and within the available space. It may be better for the programmer to indicate explicitly that an array should be single threaded, rather than leave it to the vagaries of an optimising compiler. Again, a number of researchers have proposed techniques for indicating that an array is single threaded. Most of these techniques are based on type systems [6, 19, 22]. This area seems promising, although the complexities of these type systems remain formidable. The following section presents another way of indicating explicitly the intention that an array be single threaded. Naturally, it is based on monads. The advantage of this method is that it works with existing type systems, using only the idea of an abstract data type. 4.2 Array transformers The monad of array transformers is simply the monad of state transformers, with the state taken to be an array. The definitions of M, unit, are as before. type M a = State (a, State) type State = Arr unit :: a M a unit a = λx. (a, x) ( ) :: M a (a M b) M b m k = λx. let (a, y) = m x in let (b, z) = k a y in (b, z) Previously, our state was an integer and we had an additional operation tick acting upon the state. Now our state is an array, and we have additional operations

16 corresponding to array creation, indexing, and update. block :: Val M a a block v m = let (a, x) = m (newarray v) in a fetch :: Ix M Val fetch i = λx. (index i x, x) assign :: Ix Val M () assign i v = λx. ((), update i v x) The call block v m creates a new array with all locations initialised to v, applies monad m to this initial state to yield value a and final state x, deallocates the array, and returns a. The call fetch i returns the value at index i in the current state, and leaves the state unchanged. The call assign i v returns the empty value (), and updates the state so that index i contains value v. A little thought shows that these operations are indeed single threaded. The only operation that could duplicate the array is fetch, but this may be implemented as follows: first fetch the entry at the given index in the array, and then return the pair consisting of this value and the pointer to the array. In semantic terms, fetch is strict in the array and the index, but not in the value located at the index, and assign is strict in the array and the index, but not the value assigned. (This differs from the previous section, where in order for the interpreter to be single threaded it was necessary for update to be strict in the given value. In this section, as we shall see, this strictness is removed but a spurious sequencing is introduced for evaluation of terms. In the following section, the spurious sequencing is removed, but the strictness will be reintroduced.) We may now make M into an abstract data type supporting the five operations unit,, block, fetch, and assign. The operation block plays a central role, as it is the only one that does not have M in its result type. Without block there would be no way to write a program using M that did not have M in its output type. Making M into an abstract data type guarantees that single threading is preserved, and hence it is safe to implement assignment with an in-place update. The use of data abstraction is essential for this purpose. Otherwise, one could write programs such as λx. (assign i v x, assign i w x) that violate the single threading property.

17 The interpreter may now be rewritten as follows. eval :: Term M Int eval (Var i) = fetch i eval (Con a) = unit a eval (Add t u) = eval t λa. eval u λb. unit (a + b) exec :: Comm M () exec (Asgn i t) = eval t λa. assign i a exec (Seq c d) = exec c λ(). exec d λ(). unit () exec (If t c d) = eval t λa. if a = 0 then exec c else exec d elab :: Prog Int elab (Prog c t) = block 0 (exec c λ(). eval t λa. unit a) The types show that evaluation of a term returns an integer and may access or modify the state, and that execution of a term returns nothing and may access or modify the state. In fact, evaluation only accesses the state and never alters it we will consider shortly a more refined system that allows us to indicate this. The abstract data type for M guarantees that it is safe to perform updates in place no special analysis technique is required. It is easy to see how the monad interpreter can be derived from the original, and (using the definitions given earlier) the proof of their equivalence is straightforward. The rewritten interpreter is slightly longer than the previous version, but perhaps slightly easier to read. For instance, execution of (Seq c d) can be read: compute the execution of c, then compute the execution of d, then return nothing. Compare this with the previous version, which has the unnerving property that exec d appears to the left of exec c. One drawback of this program is that it introduces too much sequencing. Evaluation of (Add t u) can be read: compute the evaluation of t, bind a to the result, then compute the evaluation of u, bind b to the result, then return a + b. This is unfortunate, in that it imposes a spurious ordering on the evaluation of t and u that was not present in the original program. The order does not matter because although eval depends on the state, it does not change it. To remedy this we will augment the monad of state transformers M with a second monad M of state readers. 4.3 Array readers Recall that the monad of array transformers takes an initial array and returns a value and a final array. type M a = State (a, State) type State = Arr

18 The corresponding monad of array readers takes an array and returns a value. No array is returned because it is assumed identical to the original array. type M a = State a unit unit a :: a M a = λx. a ( ) :: M a (a M b) M b m k = λx. let a = m x in k a x fetch fetch i :: Ix M Val = λx. index i x The call unit a ignores the given state x and returns a. The call m k performs computation m in the given state x, yielding value a, then performs computation k a in the same state x. Thus, unit discards the state and duplicates it. The call fetch i returns the value in the given state x at index i. Clearly, computations that only read the state are a subset of the computations that may read and write the state. Hence there should be a way to coerce a computation in monad M into one in monad M. coerce :: M a M a coerce m = λx. let a = m x in (a, x) The call coerce m performs computation m in the initial state x, yielding a, and returns a paired with state x. The function coerce enjoys a number of mathematical properties to be discussed shortly. Again, these operations maintain single threading if suitably implemented. The definitions of and coerce must both be strict in the intermediate value a. This guarantees that when coerce m is performed in state x, the computation of m x will reduce to a form a that contains no extant pointers to the state x before the pair (a, x) is returned. Hence there will be only one pointer extant to the state whenever it is updated. A monad is commutative if it satisfies the law m λa. n λb. o = n λb. m λa. o. The scope of a includes n on the right and not on the left, so this law is valid only when a does not appear free in n. Similarly, b must not appear free in m. In a commutative monad the order of computation does not matter. The state reader monad is commutative, while the state transfomer monad is not. So no spurious order is imposed on computations in the state reader monad. In particular, the call m k may safely be implemented so that m and k a are computed in parallel. However, the final result must still be strict in a. For instance, with the annotations used in the GRIP processor, could be defined as follows. m k = λx. let a = m x in let b = k a x in par a (par b (seq a b))

19 The two calls to par spark parallel computations of a and b, and the call to seq waits for a to reduce to a non-bottom value before returning b. These operations may be packaged into two abstract data types, M and M, supporting the eight operations unit,, unit,, block, assign, fetch, and coerce. The abstraction guarantees single threading, so assign may be implemented by an in-place update. The interpreter may be rewritten again. eval :: Term M Int eval (Var i) = fetch i eval (Con a) = unit a eval (Add t u) = eval t λa. eval u λb. unit (a + b) exec :: Comm M () exec (Asgn i t) = coerce (eval t) λa. assign i a exec (Seq c d) = exec c λ(). exec d λ(). unit () exec (If t c d) = coerce (eval t) λa. if a = 0 then exec c else exec d elab :: Prog Int elab (Prog c t) = block 0 (exec c λ(). coerce (eval t) λa. unit a) This differs from the previous version in that eval is written in terms of M rather than M, and calls to coerce surround the calls of eval in the other two functions. The new types make it clear that eval depends upon the state but does not alter it, while exec may both depend upon and alter the state. The excessive sequencing of the previous version has been eliminated. In the evaluation of (Add t u) the two subexpressions may be evaluated in either order or concurrently. A monad morphism from a monad M to a monad M is a function h :: M a M a that preserves the monad structure: h (unit a) = unit a, h (m λa. n) = (h m) λa. (h n). It often happens that one wishes to use a combination of monads to achieve a purpose, and monad morphisms play the key role of converting from one monad to another [9]. In particular, coerce is a monad morphism, and it follows immediately from this that the two versions of the interpreter are equivalent. 4.4 Conclusion How a functional language may provide in-place array update is an old problem. This section has presented a new solution, consisting of two abstract data types with eight operations between them. No change to the programming language is required, other than to provide an implementation of these types, perhaps as part of the standard prelude. The discovery of such a simple solution comes as

20 a surpise, considering the plethora of more elaborate solutions that have been proposed. A different way of expressing the same solution, based on continuation passing style, has subsequently been proposed by Hudak [8]. But Hudak s solution was inspired by the monad solution, and the monad solution still appears to have some small advantages [15]. Why was this solution not discovered twenty years ago? One possible reason is that the data types involve higher-order functions in an essential way. The usual axiomatisation of arrays involves only first-order functions, and so perhaps it did not occur to anyone to search for an abstract data type based on higher-order functions. That monads led to the discovery of the solution must count as a point in their favour. 5 Parsers Parsers are the great success story of theoretical computing. The BNF formalism provides a concise and precise way to describe the syntax of a programming language. Mathematical tests can determine if a BNF grammar is ambiguous or vacuous. Transformations can produce an equivalent grammar that is easier to parse. Compiler-compilers can turn a high-level specification of a grammar into an efficient program. This section shows how monads provide a simple framework for constructing recursive descent parsers. This is of interest in its own right, and also because the basic structures of parsing sequencing and alternation are fundamental to all of computing. It also provides a demonstration of how monads can model backtracking (or angelic non-determinism). 5.1 Lists Our representation of parsers depends upon lists. Lists are ubiquitous in functional programming, and it is surprising that we have managed to get by so far while barely mentioning them. Actually, they have appeared in disguise, as strings are simply lists of characters. We review some notation. We write [a] for the type of a list with elements all of type a, and : for cons. Thus [1, 2, 3 ] = 1 : 2 : 3 : [ ], and both have type [Int]. Strings are lists of characters, so String and [Char] are equivalent, and monad is just an abbreviation for [ m, o, n, a, d ]. It is perhaps not suprising that lists form a monad. unit unit a :: a [a] = [a] ( ) :: [a] (a [b]) [b] [ ] k = [ ] (a : x) k = k a + (x k)

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